Search

We generalize the recent invariant polytope algorithm for computing the joint spectral radius and extend it to a wider class of matrix sets. This, in particular, makes the algorithm applicable to sets of matrices that have finitely many spectrum maximizing products. A criterion of convergence of the algorithm is proved. As an application we solve two challenging computational open problems. First we find the regularity of the Butterfly subdivision scheme for various parameters $\omega$. In the “most regular” case $\omega = \frac{1}{16}$, we prove that the limit function has Holder exponent 2 and its derivative is “almost Lipschitz” with logarithmic factor 2. Second we compute the Holder exponent of Daubechies wavelets of high order.

International audience; The objective of this paper is to propose an efficient model-based bit allocation process optimizing the performances of a wavelet coder for semiregular meshes. More precisely, this process should compute the best quantizers for the wavelet coefficient subbands that minimize the reconstructed mean square error for one specific target bitrate. In order to design a fast and low complex allocation process, we propose an approximation of the reconstructed mean square error relative to the coding of semiregular mesh geometry. This error is expressed directly from the quantization errors of each coefficient subband. For that purpose, we have to take into account the influence of the wavelet filters on the quantized coefficients. Furthermore, we propose a specific approximation for wavelet transforms based on lifting schemes. Experimentally, we show that, in comparison with a “naive” approximation (depending on the subband levels), using the proposed approximation as distortion criterion during the model-based allocation process improves the performances of a wavelet-based coder for any model, any bitrate, and any lifting scheme.